Influence of attack angle on vortex-induced vibration and energy harvesting of two cylinders in side-by-side arrangement

Abstract

The vortex-induced vibration and energy harvesting of two cylinders in side-by-side arrangement with different attack angles are numerically investigated using two-dimensional unsteady Reynolds-Averaged Navier–Stokes simulations. The Reynolds number ranges from 1000 to 10,000, and the attack angle of free flow is varied from 0° to 90°. Results indicate that the vortex-induced vibration responses with attack angle range of 0°≤ α ≤ 30° are stronger than other attack angle cases. The parallel vortex streets are clearly observed with synchronized vortex shedding. Relatively large attack angle leads to a phase difference between the wake patterns of the two cylinders. Hydrokinetic energy can be obviously harvested when Re > 4000. Compared with the larger attack angle case, the two side-by-side cylinders with smaller attack angle have better performance on energy conversion. The maximum energy conversion efficiency of 21.7% is achieved. The optimum region for energy conversion is 5000 ≤ Re ≤ 7000 and 0°≤ α ≤ 30°.

Introduction

Vortex-induced vibration (VIV) is widely concerned with the rapid development of ocean engineering, wind engineering, aerospace, and nuclear engineering. As a common physical phenomenon, the flow past a bluff body and the oscillations induced by the vortex shedding have attracted a widespread attention during the past several decades, not only in theory but also in the engineering applications of the hydrodynamics. Rich fundamental fluid mechanics have been revealed.^1–4 VIV is typically treated as a destructive phenomenon because the fatigue damage may be potentially introduced. Unlike previous efforts to alter vortex shedding and suppress the occurrence of VIV, Lee et al. ⁵ and Bernitsas et al.^6,7 have successfully utilized this potentially disastrous phenomenon to generate power with the VIVACE (Vortex-Induced Vibration for Aquatic Clean Energy) converter from ocean and river. The simplest oscillatory system of the VIVACE converter consists of a rigid circular cylinder mounted on end linear springs. A power take-off system is connected to the oscillatory system by a transmission mechanism. ⁸ Abujarad et al. ⁹ reviewed researches on integration of intermittent renewable energy resources into the power networks. With increasing attention in the field of energy harvesting from ocean and river, higher energy density and efficiency are demanded. In order to enhance the hydrokinetic energy harvesting of the VIVACE converter, multiple circular cylinders are developed for the oscillatory system. The present study aims at analyzing the VIV responses of two side-by-side cylinders and achieving the best efficiency of energy conversion. The characteristics of flow around multiple cylinders are more complex than that of isolated cylinder, because of the interaction between each cylinder. The arrangement of cylinders has obvious influence on the vortex shedding, the wake patterns, and the characteristics of body motions. Amount of researches on multi-cylinder system has been reported.^10,11

For VIV of an isolated cylinder, the vortex shedding mode is closely correlated to the vibration of the cylinder. The Williamson–Roshko wake mode maps were established based on a large number of test results. Various wake pattern modes can be obtained for a rigid cylinder with low mass and low damping, including the 2S, 2P, P + S, 2P + 2S, and so on.^12–15

For VIV of two-cylinder system, there is a partition method of the near-wake structures as well. Zdravkovich and colleagues^16,17 found that the near-wake structure varies with different center-to-center distance ratio d/D of two tandem cylinders: the downstream cylinder is affected by near-wake of the upstream cylinder at d/D < 4. For the side-by-side configuration, three main wake patterns were observed, including single-bluff-body behavior, biased flow pattern, and parallel vortex streets. Different kinds of wake patterns are affected by Reynolds number, the center-to-center distance, and the experimental conditions. At comparatively low Reynolds number and small center-to-center distance ratio T/D (1 < T/D < 1.1–1.2), the wake pattern is single-bluff-body behavior. A single vortex street forms in the combined wake.^18–20 The drag of both two cylinders is reduced, the vortex of each cylinder couples, and the vortices are shed from the outside shear layer only. In this condition, the two side-by-side cylinders can be treated as the isolated cylinder. At relatively high Reynolds number and large distance ratio T/D (T/D > 2–2.5), the parallel vortex streets can be observed. The two cylinders have the same frequency of vortex shedding while oscillating independently.^21–23 It should be noticed that the parallel vortex streets are shown to be synchronized as in-phase or anti-phase due to the specific distance between the two cylinders. The biased flow pattern can be observed at intermediate distance ratio T/D when Reynolds number ranges from 1000 to 3000. ¹⁷ In biased flow pattern, the near-wake flow is unsymmetrical. One cylinder has wider near-wake, while the other cylinder has narrow near-wake.^24–26 For different Reynolds numbers and center-to-center distance ratios, the frequency of vortex shedding of the two cylinders shows different characteristics.

Apart from Reynolds number and the center-to-center distance ratio, the direction of flow also has significant influence on the VIV of two side-by-side cylinders. In engineering applications, the direction of free stream is changeable. The direction of free stream has significant effect on the vortex shedding and wake trajectory of the two cylinders. The research on this topic is still in the stage of exploration and needs to be intensively studied. In this article, the effect of attack angle on the VIV of two circular cylinders in side-by-side arrangement is numerically investigated. The results can provide additional information for understanding the mechanism of side-by-side configuration and improving the adaptation of two-cylinder system in engineering practice.

In the present study, the VIV and energy harvesting of two cylinders in side-by-side arrangement are studied using two-dimensional Unsteady Reynolds-Averaged Navier–Stokes (2-D URANS) simulations. The objective of this study is to investigate the influence of incoming flow condition on the cylinder dynamics and improve the energy harvesting efficiency. The physical model and running parameters are presented in “Physical model” section. In “Governing equations” section and “Mathematical model for energy harvesting” section, the numerical approach is described. Boundary conditions and grid requirement are presented in “Computational domain and grid generation” section. The simulation results of amplitude responses, frequency responses, near-wake structures, and energy conversion for the two cylinders are shown in “Results and discussion” section. Conclusions are presented at the end.

Physical model

The physical model considered in this article consists of an oscillatory systems mounted on end linear springs as shown in Figure 1. The elements of the oscillatory system are two rigid circular cylinders of diameter D. K is the stiffness of the supporting linear springs, and the system damping C_system due to friction (Table 1 and Table 2). In the present study, the two circular cylinders are rigidly connected with each other. There is no relative displacement between the two side-by-side cylinders when VIV occurs. The two cylinders are constrained to oscillate in a single degree of freedom motion in the y-direction only. The center-to-center distance of the two cylinders, T, is fixed at 2D. In this study, the system parameters in the numerical simulation are the same as those used in the corresponding experiments by Khalak and Williamson. ²⁷ The system parameters of the model in the simulation are listed in Table 2.

Figure 1.

Schematic of the physical model.

Table 1.

Nomenclature.

ARMS	Root-mean-square amplitude
Csystem	Total damping of system
Cstructure	Structural damping
Charness	Added damping to harness energy
D	Cylinder diameter
$f_{n,\mathrm{water}}=\sqrt{K/(m_{\mathrm{osc}}+m_a)}/2\pi$	System natural frequency in water
fosc	Oscillating frequency of cylinder
K	Spring constant
Ca	Added mass coefficient
Cd	Drag coefficient
md	Displaced fluid mass
ma = Camd	Added mass
mosc	Oscillating system mass
m* = mosc/md	Mass ratio
Re = UD/υ	Reynolds number
T	Center-to-center distance of the two side-by-side cylinders
U	Flow velocity
$U_{\mathrm{water}}^{*}=U/\left(f_{n,\mathrm{water}}D\right)$	Reduced velocity in water
α	Attack angle of flow velocity
ζ	Damping ratio of system
µt	Turbulent eddy viscosity
υ	Kinematic molecular viscosity
ρ	Density of the fluid

Table 2.

Physical model parameters.

Item		Values
Diameter of the cylinder	D (m)	0.0381
Oscillating system mass per unit length	mosc (kg/m)	2.735
Spring constant per unit length	K (N/m)	66.438
Damping ratio of system	ζ	0.0254
Damping	Csystem (Ns/m)	0.066
Natural frequency in water	fn,water	0.659
Mass ratio	m*	2.4
Density of water	ρ (kg/m3)	999.103
Kinematic molecular viscosity	υ (m2/s)	1.1389 × 10−6

Governing equations

In this study, the flow is assumed to be 2D unsteady incompressible flow. The flow is modeled using the2D-URANS equations together with the Spalart–Almaras (S-A) one-equation turbulence model. As a one-equation model, the S-A model solves for one transport equation for eddy viscosity like variable $\tilde{\upsilon }$ . The transport equation of S-A model is assembled by using empiricism and arguments of dimensional analysis. Different from other one-equation models, the S-A model is a local model, which means that the equation of one location is not influenced by the solutions of other points. Therefore, the S-A model is compatible with grids of structures, in terms of near-wall resolution and stiffness. Comparing with other turbulence models that are suffering from excessive sensitivity to free stream values, it is relatively trivial to prescribe the free stream and the wall boundary conditions of the S-A model. Additional definitions of the functions and constants are provided by Spalart and Almaras. ²⁸ The solver used in the present study is successfully applied in the previous work for VIV of multiple cylinders. The simulation results match well with the experiment. Therefore, the flow solver was chosen for the study of two side-by-side cylinders in VIV. ¹⁰

The basic URANS equations are

$$\frac{\partial U_i}{\partial x_i}=0$$

(1)

$$\frac{\partial U_i}{\partial t}+\frac{\partial }{\partial x_j}\left(U_i{U}_j\right)=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}(2\upsilon {\mathbf{S}}_{\mathrm{ij}}-\overline{u_i\prime u_j\prime })$$

(2)

where υ is the molecular kinematic viscosity and S _ij is the mean strain-rate tensor

$${\mathbf{S}}_{\mathrm{ij}}=\frac{1}{2}(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i})$$

(3)

where U_i is the mean-flow velocity vector and ${\tau }_{\mathrm{ij}}=-\rho \overline{u_j\prime u_i\prime }$ is the Reynolds-stress tensor.

The Boussinesq eddy-viscosity approximation is employed to solve the URANS equations for the mean-flow properties of the turbulence flow

$${\tau }_{\mathrm{ij}}=-\rho \overline{u_j\prime u_i\prime }=2{\mu }_t{S}_{\mathrm{ij}}-2/3k{\delta }_{\mathrm{ij}}$$

(4)

where ${\mu }_t$ is the turbulence eddy viscosity, and k is the fluctuating kinematic energy, which is defined as

$$k=\frac{1}{2}\overline{u_j\prime u_i\prime }$$

(5)

Since that the fluctuating kinematic energy k is ignored in the S-A model, the relationship between Reynolds stress and the mean velocity gradient is

$$-\rho \overline{u_i\prime u_j\prime }=2{\mu }_t{\mathbf{S}}_{\mathrm{ij}}$$

(6)

where µ_t is turbulence eddy viscosity.

In the S-A model, the turbulent eddy viscosity is computed from

$${\mu }_t=\rho \tilde{\upsilon }{f}_{\upsilon 1}$$

(7)

where

$$f_{\upsilon 1}=\frac{{\chi }^3}{{\chi }^3+{\mathrm{c}}_{\upsilon 1}^3}$$

(8)

$$\chi =\frac{\tilde{\upsilon }}{\upsilon }$$

(9)

where $\tilde{\upsilon }$ is an intermediate working variable of the turbulence model.

The dynamic of the cylinders is simplified as the classical linear oscillator system using the mass-spring-damper oscillator model

$$m\stackrel{··}{y}+C_{\mathrm{system}}\stackrel{·}{y}+\mathrm{Ky}=F_{\mathrm{fluid},y}$$

(10)

where $F_{\mathrm{fluid},y}$ is the applied force, m is the total mass of the oscillating system, C_system is the structural damping coefficient, and K is the stiffness of the spring

$$C_{\mathrm{system}}=C_{\mathrm{structure}}+C_{\mathrm{harness}}$$

(11)

where C_structure is the structural damping and C_harness is defined as the added damping to harness energy. The value of system damping is referred to our earlier work. ⁸

In this study, the simulation is performed using a solver built on the open source computational fluid dynamics (CFD) tool OpenFOAM. As a collection of C++ libraries, OpenFOAM is developed to solve continuum flow mechanics problems with the finite volume discretization procedure. A control volume integral of spatial derivative terms can be converted to a cell surface integral by Gauss’s theorem. The divergence, gradient, and the Laplacian terms in the governing equations are integrated over the control volume based on the Gauss integration using second-order linear interpolation method in space. The second-order backward Euler integration is used in the time integration. A pressure implicit with splitting of operators (PISO) algorithm is used for solving momentum and continuity equations together in a segregated way. Automatic time step is employed in the present study. To achieve temporal accuracy and numerical stability, the Courant number, Co = Δt|U|/Δx (where Δt is the time step and Δx is the cell size), less than 1 is required. Since the surface-tracking algorithm is considerably more sensitive than the standard flow calculations, an upper limit Co = 0.5 is considered in this work. The oscillator equation is two-way coupled and strong coupling algorithm is used. In one time step, the computational process is shown in Figure 2.

Figure 2.

Computational process in one time step.

Mathematical model for energy harvesting

In this section, the mass-spring-damper oscillator model of the oscillator system in equation (8) is used to calculate the harnessed energy. The power generated from the flow via the vibrating cylinders per oscillation cycle is given as

$$P_{\mathrm{mech}}=\frac{1}{T_{\mathrm{osc}}}\underset{0}{\overset{T_{\mathrm{osc}}}{\int }}{F}_{\mathrm{fluid},y}\mathrm{dt}$$

(12)

Plugging equation (10) into equation (12), the power can be rewritten as

$$P_{\mathrm{mech}}=\frac{1}{T_{\mathrm{osc}}}\underset{0}{\overset{T_{\mathrm{osc}}}{\int }}(m\stackrel{··}{y}+C_{\mathrm{system}}\stackrel{·}{y}+\mathrm{Ky})\mathrm{dt}$$

(13)

Since the VIV responses of rigid circular cylinders are assumed to be nearly sinusoidal,^8,10 the displacement of the cylinders is

$$y=A\sin \left({\omega }_{\mathrm{osc}}t\right)$$

(14)

where A is the amplitude, and ${\omega }_{\mathrm{osc}}$ is the angular frequency, which is defined as

$${\omega }_{\mathrm{osc}}=2\pi f_{\mathrm{osc}}$$

(15)

The velocity and acceleration of the cylinders in the y-direction are

$$\stackrel{·}{y}=2\pi f_{\mathrm{osc}}A\cos \left({\omega }_{\mathrm{osc}}t\right)$$

(16)

$$\stackrel{··}{y}=-4{\pi }^2{f}_{\mathrm{osc}}^{2}A\sin \left({\omega }_{\mathrm{osc}}t\right)$$

(17)

Inserting equations (14) to (17) into equation (13), the power in the oscillation system can be expressed as

$$P_{\mathrm{mech}}=2{\pi }^2{f}_{\mathrm{osc}}^2{A}^2{C}_{\mathrm{system}}$$

(18)

The elastic power of the oscillation system is

$$P_k=\frac{1}{T_{\mathrm{osc}}}\underset{0}{\overset{A}{\int }}{F}_k\mathrm{dy}=\frac{1}{2}K{A}^2{f}_{\mathrm{osc}}$$

(19)

Thus, the converted power is

$$P_{\mathrm{convert}}=P_{\mathrm{mech}}+P_k=2{\pi }^2{f}_{\mathrm{osc}}^2{A}^2{C}_{\mathrm{system}}+\frac{1}{2}K{A}^2{f}_{\mathrm{osc}}$$

(20)

Based on the Bernoulli equation, the kinetic pressure head is defined as

$$P_T=\frac{1}{2}\rho U^2$$

(21)

The fluid force acting on the cylinder is

$$F_{\chi }=\frac{1}{2}\rho U^2\mathrm{DL}$$

(22)

The power in the fluid can be calculated as

$$P_{\mathrm{fluid}}=F_{\chi }U=\frac{1}{2}\rho U^3\mathrm{DL}$$

(23)

The power ratio is defined as the ratio of converted power to the power in the fluid, which can be written as

$${\eta }_{\mathrm{convert}}=\frac{P_{\mathrm{convert}}}{P_{\mathrm{fluid}}}$$

(24)

Computational domain and grid generation

As shown in Figure 3(a), the computational domain in the present work is a circle with diameter of 50D. The two side-by-side cylinders are located in the center of the computational domain. The computational domain can be divided into the upstream and downstream subdomains by the straight line which is perpendicular to the flow velocity direction and passes through the domain center. The boundary of the upstream is the inflow and the boundary of the downstream is the outflow. The inflow and outflow boundaries vary with the attack angle of the free stream velocity. The inflow velocity is considered as constant and uniform. A zero gradient condition is specified for velocity at the outflow boundary. When the cylinders undergo VIV, a moving wall boundary condition is applied for the two cylinders.

Figure 3.

(a) Computational domain and (b) close-up of the grid for two side-by-side cylinders.

The flow field is meshed in structured grid with local grid refinement as illustrated in Figure 3(b). The attack angle, α, which is defined as the angle between the x-direction and the free stream, is varied from 0° to 90°. The simulations are conducted in Reynolds number ranging from 1000 to 10,000.

In order to determine the overall grid independence to achieve a convergent and accurate solution in reasonable computational time, a grid sensitivity study was conducted on three different grid levels. The three grids are named based on the number of cells, which increases with a factor of approximately 1.5 in circumferential and radial direction. The detailed parameters of grid independence study are listed in Table 3. The flow parameters were calculated for comparison, where C_D and C_L are the time-average value of the drag coefficient and the lift coefficient. In the present study, all three grids obtain similar results. In view of the computer resources and computing time, cell numbers of 80,400 were chosen for all cases. Based on the turbulence model and wall functions which are used for simulating the VIV of the two cylinders, the first grid spacing y+ about 30–70 is employed in this study.

Table 3.

Grid independence study (Re = 2000, α = 5°).

Grid	CD	CL
Coarse (46,240)	2.013	0.238
Medium (80,400)	2.039	0.247
Fine (111,680)	2.051	0.249

Results and discussion

In this section, VIV of the two side-by-side cylinders at different attack angles is numerically investigated. The responses of amplitude, frequency, wake patterns, and energy conversion of cylinders with different attack angles are discussed.

Amplitude responses

In this section, the amplitude responses of the two circular cylinders in side-by-side arrangement are discussed. The traditional measurement of VIV response is the peak amplitude of oscillation. This measurement admits harmonic body oscillations only, which means that the peak amplitude and frequency of the oscillating cylinders are required to fully describe the motion. In case that the flow is not completely synchronized to the body motions, the root-mean-square amplitude can provide a more meaningful measure of magnitude of asymmetric oscillation. Instead of the peak amplitude ratio, the variation of root-mean-square amplitude ratio (A_RMS/D) with respect to the attack angle and Reynolds number is presented in Figure 4. Since the Reynolds number range is 1000 ≤Re ≤ 10,000, the corresponding reduced velocity range is 1.19 ≤  $U_{\mathrm{water}}^{*}$  ≤11.90.

1. $U_{\mathrm{water}}^{*}$  = 1.19 (Re = 1000): No obvious VIV is observed. In this region, the amplitude ratio is in the order of 10⁻³, while the maximum amplitude is 0.0023D at α = 70°.

2. 2.38 ≤  $U_{\mathrm{water}}^{*}$  ≤ 3.57 (2000 ≤ Re ≤ 3000): The two side-by-side cylinders start to oscillate at $U_{\mathrm{water}}^{*}$  = 2.38 (Re = 2000) and VIV can be captured. The VIV initial branch is obtained. The amplitude decreases continuously with the attack angle increasing from 0° to 70°. Particularly, a sharp decrease is observed when the attack angle rises from 10° to 20°. The minimum amplitude of 0.0089D and 0.019D for Re = 2000 and Re = 3000 is observed, respectively. The amplitude increases in the range of 70°≤ α ≤ 90°. The amplitude reaches the peak of 0.019D for $U_{\mathrm{water}}^{*}$  = 2.38 (Re = 2000) at α = 90°, and 0.097D for $U_{\mathrm{water}}^{*}$  = 3.57 (Re = 3000) at α = 0°.

3. 3.57 <  $U_{\mathrm{water}}^{*}$  ≤ 7.14 (3000 < Re ≤ 6000): The amplitude decreases sharply at α = 5°, then climbs to the peak at α = 10°. The peak amplitude is 0.20D for $U_{\mathrm{water}}^{*}$  = 4.76 (Re = 4000), 0.30D for $U_{\mathrm{water}}^{*}$  = 5.95 (Re = 5000), and 0.31D for $U_{\mathrm{water}}^{*}$  = 7.14 (Re = 6000), respectively. The upper branch is observed. When 20°≤ α ≤ 70°, the amplitude continues to decrease, which is similar to the cases of $U_{\mathrm{water}}^{*}$  = 2.38 (Re = 2000) and $U_{\mathrm{water}}^{*}$  = 3.57 (Re = 3000). The amplitude reaches the bottom at α = 70°.

4. 7.14 <  $U_{\mathrm{water}}^{*}$  ≤ 11.90 (6000 < Re ≤ 10,000): A sharp drop of amplitude is observed when the attack angle reaches 5°. Then a local peak is obtained when the attack angle approaches 10°. The amplitude shows a decrease with the attack angle rising from 20° to 70°, then a slight growth is captured as the attack angle keeps growing to 90°. In this region, the VIV branch transforms to lower branch. The maximum value of the amplitude is observed at α = 0°. The maximum value of amplitude is 0.29D for $U_{\mathrm{water}}^{*}$  = 8.33 (Re = 7000), 0.27D for $U_{\mathrm{water}}^{*}$  = 9.52 (Re = 8000), 0.24D for $U_{\mathrm{water}}^{*}$  = 10.71 (Re = 9000), and 0.27D for $U_{\mathrm{water}}^{*}$  = 11.90 (Re = 10,000), respectively.

Figure 4.

Amplitude ratios of the cylinders in VIV.

From above, the amplitude ratio increases significantly as the reduced velocity rises. Relatively high velocity leads to comparatively complex vortex street interaction in the wake of the two cylinders. In addition, the variation range of the amplitude for 4.76 ≤  $U_{\mathrm{water}}^{*}$  ≤ 7.14 (4000 ≤ Re ≤ 6000) is higher than other cases. The strongest oscillation is obtained when $U_{\mathrm{water}}^{*}$  = 7.14 (Re = 6000) at α = 10°, where the amplitude reaches the maximum value of 0.31D. The minimum amplitude of the two side-by-side cylinders is observed at α = 70°. When the attack angle is small (α ≤ 10°), the amplitude rises sharply with reduced velocity increasing to 5.95 (Re = 5000), then fluctuates in the range of 0.20D ∼ 0.30D. When 20°≤ α ≤ 70°, the oscillation of the upper cylinder is significantly influenced by the lower cylinder with increasing attack angle, resulting a continuous decrease of the amplitude. When the attack angle approaches 90°, the upper cylinder is completely in the wake of the lower cylinder. For Reynolds number less than 10,000, a typical VIV response consists of the initial branch, the upper branch with a constant amplitude, and the lower branch. ²⁹ For cases of higher Reynolds numbers, a new branch is observed that the amplitude increases with the increase of Re.

Frequency responses

The simulation records for the two side-by-side cylinders are processed using Fast Fourier Transform (FFT). The frequency of oscillation is calculated and plotted versus the attack angle in Figure 5. The frequency of oscillation (f_osc) for the two cylinders is non-dimensionalized by the corresponding system natural frequency in water, f_n,water.

Figure 5.

Frequency ratios of the cylinders in VIV.

$U_{\mathrm{water}}^{*}$  = 1.19 (Re = 1000): In the range of 30°≤ α ≤ 90°, the frequency ratio generally decreases with the increasing attack angle. No VIV takes place in this region. However, when α ≤ 30° the frequency ratio oscillates between 0.2 and 0.3. The maximum frequency ratio is 0.32 at α = 30°.

1.19 <  $U_{\mathrm{water}}^{*}$  < 2.38 (1000 < Re < 2000): In this range, there is a dramatically increase of frequency ratio. The cylinders start to oscillate and the major harmonic frequency in the VIV initial branch is obtained.

1.19 <  $U_{\mathrm{water}}^{*}$  ≤ 7.14 (1000 < Re ≤ 6000): When the velocity increases from 1.19 (Re = 1000) to 2.38 (Re = 2000), there is a dramatically increase of frequency ratio. When the velocity is greater than 2.38, it follows the similar variation trend with the case of $U_{\mathrm{water}}^{*}$  = 1.19 (Re = 1000). The maximum frequency ratio is 1.44 at α = 90°.

7.14 <  $U_{\mathrm{water}}^{*}$  ≤ 10.91 (6000 < Re ≤ 9000): A slight decrease of the frequency ratio is observed in the variation of 0°≤ α ≤ 5°. After a relatively slow rate of increasing, a prominent drop occurs separately when α rises to 50°. Unlike the cases of Re = 1000 and Re = 2000, the frequency ratio is minimized at α = 90° with a value of 1.14.

10.91 <  $U_{\mathrm{water}}^{*}$  ≤ 11.9 (9000 < Re ≤ 10,000): In this range, the variation trend does not change obviously. It should be mentioned that the frequency ratio shows the greatest decrease when 20°≤ α ≤ 30°.

From the discussion above, the vibration of the two cylinders is almost not initiated when Re = 1000. Comparatively, low-frequency ratios of the two cylinders are obtained. The frequency ratios are fairly close in the variation of 2000 ≤ Re ≤ 10,000 and stay around 1.1. The frequency ratio reaches the highest value of 1.33 at Re = 4000 and α = 90°.

Near-wake structures

In this section, the simulation results of vortex patterns are discussed. For the two circular cylinders in side-by-side arrangement, the wakes of the two cylinders interact with one another on either side of the gap between the two cylinders, which results in more complex vortex structure than that for the isolated circular cylinder. The vortex shedding patterns at different attack angles of flow for Re = 2000, Re = 4000, Re = 6000, and Re = 8000 are presented in Figures 6–9, respectively.

Figure 6.

Vortex structures of the two cylinders in side-by-side arrangement at different attack angles of flow for Re = 2000.

Figure 7.

Vortex structures of the two cylinders in side-by-side arrangement at different attack angles of flow for Re = 4000.

Figure 8.

Vortex structures of the two cylinders in side-by-side arrangement at different attack angles of flow for Re = 6000.

Figure 9.

Vortex structures of the two cylinders in side-by-side arrangement at different attack angles of flow for Re = 8000.

$U_{\mathrm{water}}^{*}$  = 2.38 (Re = 2000): The vortex structures of the two cylinders in side-by-side arrangement at different attack angles are shown in Figure 6. The vortices are numbered as V1 to V4 in the order of their formation and shedding in the near-wake. From the discussion of amplitude responses above, small displacement of the two side-by-side cylinders is achieved due to the relatively low flow velocity, and no obvious VIV occurs in this velocity range. The vibration of the two cylinders is almost not initiated. As presented in Figure 6(a)–(e), the typical 2S mode is clearly observed for each cylinder. Here, 2S indicates two single vortices shed per cycle. In the combined wake, the parallel vortex streets can be captured obviously. Specifically, as seen in Figure 6(a), the clockwise rotating vortex V1 is shed from the top shear layer of the upper cylinder with the upward cylinder motion. Meanwhile, the counter-clockwise rotating vortex V2 is shed from the bottom shear layer of the lower cylinder. Vortices V3 and V4 are shed simultaneously in the near-wake during the downward journey. The instantaneous wake shows symmetric configuration with regard to the wake centerline at α = 0°. In this condition, the flow gains its symmetry. An anti-phase synchronized vortex shedding is formed individually behind the two side-by-side cylinders. The vortices shed from the two cylinders have same intensity and opposite directions. The resultant force acted on the two cylinders is almost equal to zero because of the opposite lift direction of each cylinder. This phenomenon leads to the inhibition of oscillation. From Figure 6(b)–(e), the vortex shedding of each cylinder is not considerably affected by the other cylinder in the range of 5°≤ α ≤ 30°. As illustrated in Figure 6(e) and (f), it is obviously observed that the vortex formation of the lower cylinder is influenced by the existence of the upper cylinder when the attack angle achieves 50°. In the upward journey, vortices V1 and V2 are formed and shed from the two cylinders. The top shear layer from the upper cylinder is cut and generated V3 in the progress of the two shed vortices moving downstream. V4 and V5 grow up during the downward cylinder motion. The vortex shedding of the upper cylinder is obviously disrupted by the shed vortices from the lower cylinder. When the attack angle reaches 90°, the upper cylinder is completely in the region of the wake behind the lower cylinder. As shown in Figure 6(h), vortices V1 and V2 with the opposite direction of rotation are generated from the upper cylinder. The shear layer of the lower cylinder is elongated and no vortex shed from the lower cylinder.

$U_{\mathrm{water}}^{*}$  = 4.76 (Re = 4000): When the flow velocity direction is perpendicular to the y-direction, the parallel vortex streets are clearly captured (see Figure 7(a)). An anti-phase synchronized vortex shedding is formed individually behind the two side-by-side cylinders. The positive and negative vortices alternately form in the wake, exhibiting a typical 2S mode structure. With the attack angle varying from 0° to 30°, vortex patterns are similar to the case of Re = 2000 due to the small impact of vortex on the motions of the two side-by-side cylinders. When the attack angle reaches 50°, the motions and the vortex shedding of the upper cylinder are affected by the lower cylinder. At α = 70° as shown in Figure 7(g), the near-wake structures of two cylinders at Re = 4000 are different from the structures when Re = 2000. In the upward journey, the shear layer of the lower cylinder is elongated to the upper cylinder, generating vortices V1 and V2. Vortex V4 is generated along with V3 as the two cylinders traverse down. The clockwise rotating vortex V1 and the counter-clockwise vortex V2 move downstream into the domain of the spacing between the two cylinders for large attack angle cases. Due to the specific spacing between the two cylinders, the vortices from the lower cylinder start to absorb the same rotation vortices from the upper cylinder. Only small-scale and very weak vortices are allowed to shed from shear layer of the upper cylinder. The suppression of the vortex structure of the upper cylinder is clearly observed in Figure 7(f) and (g). When the attack angle approaches 90°, the vortices formed from the lower cylinder can be shed successfully. Two vortices with the opposite direction of rotation, V1 from the lower cylinder and V2 from the upper cylinder, are shed first during the gap when the two cylinders move upward. Vortex V3 is conducted thereafter. V4 and V5 are generated in the downward travel.

$U_{\mathrm{water}}^{*}$  = 7.14 (Re = 6000): Figure 8 presents the vortex patterns at different attack angles for Re = 6000. As illustrated in Figure 8(a), the flow patterns of two cylinders are symmetrical approximately. Relatively high Reynolds number leads to comparatively complex vortex street interaction in the combined wake of the cylinders. An anti-phase synchronized vortex shedding with the same-signed vortices shed in the near-wake is shown in the wake region. In the progress of vortex shedding from the two cylinders, the lift force of each cylinder is in the same direction. 2P mode is clearly observed for each cylinder. Here, 2P indicates two pairs of vortices shed per cycle. From Figure 8(b)–(e), the wake pattern becomes asymmetrical, characterized by a gap flow biased toward the upper cylinder. The near-wake of the lower cylinder is wider than the wake of the upper cylinder. The biased flow pattern leads to a phase difference as well. When the two cylinders move in phase, the vortex structure of each cylinder is P + S. The vortices shed from the lower cylinder directly and closely interact with the upper cylinder. The maximum value of amplitude is obtained. The motion of the cylinders shows a periodic pattern, while regular small displacement is observed. The reason for the small displacement is that the vortex of the upper cylinder is weakened by the vortices shed from the lower cylinder. As shown in Figure 8(f), only the vortices shed from the outside shear layers of the two cylinders can be observed, since the vortices from each cylinder impact and merge with each other. With α larger than 70°, the upper cylinder stays in the wake of the lower cylinder, and the motions of the lower cylinder are inhibited by the upper cylinder due to the specific distance between the two cylinders.

$U_{\mathrm{water}}^{*}$  = 9.52 (Re = 8000): Figure 9 presents the vortex patterns at different attack angles for Re = 8000. Similar to the Re = 4000 and Re = 6000 cases, an anti-phase synchronized vortex shedding is observed. 2S mode is observed for each cylinder. The shed vortices interact and merge with each other in the progress of moving downstream, while scarcely have influence on the motions of the two cylinders (see Figure 9(a) and (d)). With attack angle increasing to 30°, a biased gap flow appears. With the attack angle rise to 50°~ 70°, the vortex shedding mode changes to P + S mode for each cylinder. Two pairs of vortices shed in the wake during the upward journey, while only two vortices shed when the cylinders move downward. When α = 90°, comparatively complex vortex patterns are observed. It is hard to identify the vortex pattern for the upper cylinder (see Figure 9(h)).

Energy conversion

This section presents the energy conversion comparison of two cylinders in side-by-side arrangement with different attack angles. Based on the responses of amplitude and frequency, the power that can be harnessed is discussed. The mathematical model for the converted power is summarized in the “Mathematical model for energy harvesting” section. Results of harnessed powers by VIV of two cylinders in side-by-side arrangement with different attack angles versus Reynolds number are shown in Figure 10. The system damping ratio for energy harvesting is ζ = 0.0254 for all cases. The system damping coefficient value per unit length is C_system = 0.066 Ns/m. Since that VIV is suppressed when damping is too high, ³⁰ relatively low mass damping is chosen for this study. The system damping calculated in this study is the same as the value for experiment of Khalak and Williamson. ²⁷

Figure 10.

Harnessed power of two cylinders in side-by-side arrangement with different attack angles.

As shown in Figure 10, when $U_{\mathrm{water}}^{*}$  ≤ 4.76 (Re ≤ 4000), the power harnessed from the fluid is close to zero since VIV of two cylinders in side-by-side arrangement is hardly initiated. As presented in equation (18), the value of converted power by VIV of the two parallel cylinders is in proportion to the square of amplitude and frequency. Only 0.024 W is achieved when α = 0° and $U_{\mathrm{water}}^{*}$  = 4.76 (Re = 4000). When Re > 4000, oscillations of the two cylinders are initiated and energy can be harvested by VIV. The differences of energy conversion with different attack angles are exhibited, resulting a steep increase in harnessed power. The power can be harnessed obviously over a broad range of Reynolds number from 5000 to 10,000, with attack angle range of 0°≤ α ≤ 30°. When the attack angle is relatively small (0°≤ α ≤ 30°), the converted power reaches a local peak and then drops off when 8.33 ≤  $U_{\mathrm{water}}^{*}$  ≤ 10.71 (7000 ≤ Re ≤ 9000) due to both amplitude and frequency decrease in this range. The maximum converted power of 0.34 W is achieved at $U_{\mathrm{water}}^{*}$  = 11.90 (Re = 10,000) when α = 20°. It should be noted that the converted power starts to fluctuate near 0.2D when Re > 6000 in the attack angle range of 20°≤ α ≤ 30°. As the attack angle rises to 50°, the converted power shows completely different trend. The harnessed power rises gradually with increasing Reynolds number. The converted powers are much lower than those smaller attack angle cases.

The energy conversion efficiencies of VIV for the two cylinders in side-by-side arrangement with different attack angles are calculated by equation (22). The comparison of power efficiencies is depicted in Figure 11. The optimum regime to harvest hydrokinetic energy is evaluated in this study. As shown in Figure 11, the maximum energy conversion efficiencies with investigated attack angles are obtained in the region of 5.95 ≤  $U_{\mathrm{water}}^{*}$  ≤ 11.90 (5000 ≤ Re ≤ 10,000). It is noteworthy that the energy conversion efficiency increases significantly with Reynolds number increasing from 1000 to 5000. For cases of comparatively small attack angles (0°≤ α ≤ 30°), the power efficiencies peak in the region of 5.95 ≤  $U_{\mathrm{water}}^{*}$  ≤ 8.33 (5000 ≤ Re ≤ 7000), then decrease with fluctuation with increasing Reynolds number. The maximum energy conversion efficiency of 21.7% is achieved at $U_{\mathrm{water}}^{*}$  = 7.14 (Re = 6000) when α = 0°. With the attack angle increasing to 50°, the power efficiency obviously drops. From the amplitude and frequency responses discussion, the simulation results show that two parallel cylinders with smaller attack angles (0°≤ α ≤ 30°) have stronger VIV responses than the larger attack angle cases. Thus, the two parallel cylinders with smaller attack angles can reach higher value of energy conversion in VIV. It can be concluded that the optimum region for energy conversion is 5000 ≤ Re ≤ 7000 and 0°≤ α ≤ 30°. Comparing with other energy harvesting converter from offshore structure such as Vertical Axis Autorotation Current Turbine (VAACT), ³¹ the energy conversion efficiency in the present study peaks at 0.22, which is lower than the efficiency 0.6 ∼ 0.7 from VAACT. In future work, to achieve better performance of energy conversion, the effect of arrangement and body shapes on VIV of multiple cylinders will be checked.

Figure 11.

Power conversion efficiency of two cylinders in side-by-side arrangement with different attack angles.

Conclusion

In this study, the VIV of two rigid circular cylinders in side-by-side arrangement with different attack angles of flow is investigated using 2-D URANS simulations with the S-A one-eq26 turbulence model. The two cylinders are mounted on end linear springs with one degree of freedom. Numerical simulations are conducted in Reynolds number range from 1000 to 10,000. The attack angle of flow is varied from 0° to 90°. Some main conclusions concerning the results are summarized as follows:

1. VIV of two side-by-side circular cylinders is obviously influenced by the attack angle of flow. The numerical tool can successfully simulate the effect of attack angle on the VIV behavior of two cylinders in side-by-side arrangement.

2. No obvious VIV is observed at Re = 1000. When 2000 ≤ Re ≤ 10,000, the minimum amplitude ratio of the two cylinders is observed at α = 70° in the varied attack angles. The VIV responses with attack angle range of 0°≤ α ≤ 30° are obviously stronger than other attack angle cases. The maximum amplitude 0.31D is achieved $U_{\mathrm{water}}^{*}$  = 7.14 (Re = 6000) at α = 10°.

3. Comparatively low-frequency ratios of the two cylinders are obtained at Re = 1000. The frequency ratios are fairly close in the variation of 2000 ≤ Re ≤ 10,000 staying around 1.1 in the investigated attack angles.

4. The vortex shedding pattern changes from one mode to another with Reynolds number corresponding to amplitude and frequency responses. The parallel vortex streets are clearly observed with synchronized vortex shedding at α = 0°. With attack angle increasing to 30°, the oscillation of the two cylinders is not fully synchronized due to the specific distance between the two cylinders, lead to a phase difference between the wake patterns of the two cylinders.

5. Hydrokinetic energy can be obviously harvested by VIV of the two cylinders in side-by-side arrangement when Re > 4000. The two parallel cylinders with smaller attack angles (0°≤ α ≤ 30°) have better performance on energy conversion than the larger attack angle cases. The maximum energy conversion efficiency of 21.7% is achieved. The optimum region for energy conversion is 5000 ≤ Re ≤ 7000 and 0°≤ α ≤ 30°.

6. In future work, the influence of the spacing on the VIV behavior and energy harvesting of the two separated cylinders in side-by-side arrangement will be studied. And two-degree-of-freedom vortex-induced vibrations of multiple cylinders will be considered as well.